Uniqueness and stability of parameter identification in elliptic boundary value problem

Abstract

This paper concerns the uniqueness and stability of an inverse problemin PDE. Our problem consists of identifying two parameters b(x)b(x) and c(x)c(x) in the following boundary-value problem {Lu:=−b(x)u′′(x)+c(x)u′(x)=f(x),u(0)=u(1)=0,{Lu:=−b(x)u″(x)+c(x)u′(x)=f(x),u(0)=u(1)=0, from distributed observations u1u1 (resp. u2u2) associated with the source f1f1 (resp. f2f2). For one observation, the solution is not unique. However, we prove, under some conditions, the uniqueness of the solution p=(b,c)p=(b,c) in the case of two observations. Furthermore, we derive a H\"older-type stability result. The algorithm of reconstruction uses the least squares method. Finally, we present some numerical examples with exact and noisy data to illustrate our method.